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El. knyga: Algebraic Topology of Finite Topological Spaces and Applications

  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2032
  • Išleidimo metai: 24-Aug-2011
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642220036
Kitos knygos pagal šią temą:
Algebraic Topology of Finite Topological Spaces and ApplicationsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2032
  • Išleidimo metai: 24-Aug-2011
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642220036
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This volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.

Recenzijos

From the reviews:

This book deals with the algebraic topology of finite topological spaces and its applications, and includes well-known results on finite spaces and original results developed by the author. The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory. (Fumihiro Ushitaki, Mathematical Reviews, March, 2014)

This book illustrates convincingly the idea that the study of finite non-Hausdorff spaces from a homotopical point of view is useful in many areas and can even be used to study well-known problems in classical algebraic topology. This book is a revised version of the PhD Thesis of the author. All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest. (Etienne Fieux, Zentralblatt MATH, Vol. 1235, 2012)

1 Preliminaries
1(18)
1.1 Finite Spaces and Posets
2(2)
1.2 Maps, Homotopies and Connectedness
4(2)
1.3 Homotopy Types
6(4)
1.4 Weak Homotopy Types: The Theory of McCord
10(9)
2 Basic Topological Properties of Finite Spaces
19(18)
2.1 Homotopy and Contiguity
19(1)
2.2 Minimal Pairs
20(2)
2.3 T1-Spaces
22(1)
2.4 Loops in the Hasse Diagram and the Fundamental Group
22(3)
2.5 Euler Characteristic
25(2)
2.6 Automorphism Groups of Finite Posets
27(2)
2.7 Joins, Products, Quotients and Wedges
29(5)
2.8 A Finite Analogue of the Mapping Cylinder
34(3)
3 Minimal Finite Models
37(12)
3.1 A Finite Space Approximation
37(2)
3.2 Minimal Finite Models of the Spheres
39(1)
3.3 Minimal Finite Models of Graphs
40(4)
3.4 The fx(X)
44(5)
4 Simple Homotopy Types and Finite Spaces
49(24)
4.1 Whitehead's Simple Homotopy Types
50(3)
4.2 Simple Homotopy Types: The First Main Theorem
53(7)
4.3 Joins, Products, Wedges and Collapsibility
60(4)
4.4 Simple Homotopy Equivalences: The Second Main Theorem
64(4)
4.5 A Simple Homotopy Version of Quillen's Theorem A
68(2)
4.6 Simple, Strong and Weak Homotopy in Two Steps
70(3)
5 Strong Homotopy Types
73(12)
5.1 A Simplicial Notion of Homotopy
73(4)
5.2 Relationship with Finite Spaces and Barycentric Subdivisions
77(3)
5.3 Nerves of Covers and the Nerve of a Complex
80(5)
6 Methods of Reduction
85(8)
6.1 Osaki's Reduction Methods
85(2)
6.2 γ-Points and One-Point Reduction Methods
87(6)
7 h-Regular Complexes and Quotients
93(12)
7.1 h-Regular CW-Complexes and Their Associated Finite Spaces
93(7)
7.2 Quotients of Finite Spaces: An Exact Sequence for Homology Groups
100(5)
8 Group Actions and a Conjecture of Quillen
105(16)
8.1 Equivariant Homotopy Theory for Finite Spaces
106(2)
8.2 The Poset of Nontrivial p-Subgroups and the Conjecture of Quillen
108(3)
8.3 Equivariant Simple Homotopy Types
111(8)
8.4 Applications to Quillen's Work
119(2)
9 Reduced Lattices
121(8)
9.1 The Homotopy of Reduced Lattices
121(3)
9.2 The L Complex
124(5)
10 Fixed Points and the Lefschetz Number
129(8)
10.1 The Fixed Point Property for Finite Spaces
129(4)
10.2 On the Lefschetz Theorem for Simplicial Automorphisms
133(4)
11 The Andrews-Curtis Conjecture
137(14)
11.1 n-Deformations and Statements of the Conjectures
137(2)
11.2 Quasi Constructible Complexes
139(6)
11.3 The Dual Notion of Quasi Constructibility
145(6)
12 Appendix
151(1)
A.1 Simplicial Complexes 151(4)
A.2 CW-Complexes and a Gluing Theorem 155(6)
References 161(4)
List of Symbols 165(2)
Index 167
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